The aim of this work is to provide a way to compute the cohomological Donaldson-Thomas invariants by localization, namely, given a d-critical scheme (or stack) \(X\)  with a \(\mathbb {C}^\ast \) -action and a \(\mathbb {C}^\ast \) -invariant d-critical structure, express the invariants of \(X\)  in terms of the invariants of the \(\mathbb {C}^\ast \) -fixed scheme (or stack) \(X^0\) . Graber and Pandharipande have proven a torus localization formula for numerical Donaldson-Thomas invariants in [1]}. An analogue formula was derived in [2]} using the alternative definition as weighted Euler numbers: the numerical Donaldson-Thomas invariants of \(X\)  are the sum of those of each components of \(X^0\)  weighted by a sign given by the parity of the dimension of the normal space to the component. Hence numerical Donaldson-Thomas invariants localize under torus action exactly like the Euler numbers of smooth spaces.
